Problem: Integrate. $\int\left(\dfrac5x+e^x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $5\ln|x|+\ln|x|+C$ (Choice B) B $\dfrac5x+e^x+C$ (Choice C) C $\dfrac5x+\ln|x|+C$ (Choice D) D $5\ln|x|+e^x+C$
Solution: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(\dfrac5x+e^x \right)dx \\\\ &=5\int \dfrac1x\,dx+\int e^x \,dx \\\\ &=5\ln|x|+e^x+C \end{aligned}$